by PutCallParrot

**Puts and Calls**

**A put is the right, but not the obligation, to sell an asset for a specific price.** The asset the put holder has a right to sell is called the **“underlying.”** The price at which the put option allows the option holder to sell is called the **“strike.”** Put options have an **expiration date**, after which they cannot be exercised, which is to say you owner of the put option can no longer invoke their right to sell after the option has expired. Now, let’s put it all together into once sentence: **When a person trades a put option, they are trading the right to sell the underlying at the strike price, subject to the eventual expiration.**

That sounds great, but how could the right to sell something at a certain price be valuable? At this point, you need to have a notion of what “market price” means. Let’s review for a moment before we dive in, using stocks as an example. When you trade a stock, you buy or sell that stock based on an agreed upon price. Traders can bid (offer to purchase) or “ask” (offer to sell) a stock at any price, but only when one trader’s bid to buy is higher than another trader’s ask to sell does a trade actually occur. That point of overlap is called the “market price.” In reality, the bid involved will end up being slightly higher than the ask, and a series of financial institutions ultimately make their money by pocketing the difference, but for our purposes, we’ll say that the market price is just where bids and asks collide. For our purposes, we may also refer to the market price as the “spot price” for now. Either term is fine for the moment.

**Puts have value because the strike price of a put may be different than the market price of the underlying.** For example, if you own a put with a strike of $5, and the underlying has a market price of $3, you have the right to buy the underlying for $3, and then turn around and exercise that put to sell the underlying to collect $5, ultimately making $2 of profit. That $2 is called the “intrinsic value” of the put option. **The intrinsic value of an option is how much you could theoretically profit at a given moment if you exercise it.**

Here is another way to look at intrinsic value: at the very moment when the option is just about to expire, and there is not enough time for the underlying price to change before expiration, the option would be worth $2. In other words, the option holder could theoretically sell their option and collect $2 instead of exercising it. Why an option holder might choose to sell rather than exercise an option will become clear as you read this series. What it comes down to is that the amount you can sell an option for may be greater than the amount you could stand to profit by exercising it. In other words, **an option may be worth more than its intrinsic value except at the exact moment of expiration**; just how is discussed later in this post.

For now though, it’s alright to be content with the notion that put options have value and that an option holder may choose to sell a put option rather than exercise it. Remember, the definition; a put is “the right, but *not* the obligation” to sell.

Now, let’s define call options, using what we have learned about puts: **A call option is the right, but not the obligation to buy (not sell) an underlying asset at a strike price, subject to expiration.** It should come as little surprise that call options also have value for reasons similar to put options. See for yourself. **Question 1:** if you had a call option with a strike of $5, and the underlying had a market price of $10, how much could you profit by exercising your call option to buy the underlying and then turning around and selling the underlying on the market? I’ll put the answer at the end of this post.

**Remember, calls let you buy at the strike. Puts let you sell at the strike.**

Like put options, call options can be traded, their intrinsic value is based on the difference between the strike price and the market price of the underlying, and they may be worth more than their intrinsic value except at the moment they expire.

Let’s recap: puts are a right to sell, and calls are a right to buy. One good way to remember it is to imagine the underlying were a dog; if you call the dog, it will come to you. Puts and calls can both be exercised or traded. The option holder gets to decide if they want to exercise or trade, but the option can’t be exercised after it expires. Depending on the price of the underlying, exercising can lead to profits, and that possibility is why options have value.

We will now go over what can cause an option to trade for more than its intrinsic value. There are a few variables that can affect options valuation. We have mentioned two so far; 1) the strike price of the option and 2) the market price of the underlying. Volatility is the first one we will go over (later in this post) because it is the easiest to understand given only the strike and the spot. In future posts, we will go over the role of the time left until expiration and the role of prevailing interest rates; specifically something called the “riskless rate,” or “rf” for short. There are other variables that can be incorporated into options pricing models, but we will not address them until we have gone over at least two models that deal with these first five.

To be clear, we are building up to being able to use 2 models that both incorporate five key variables, and you’ll be able to use 3 of the variables a simplified version of one of the two models by the end of this post. The key variables are, as previously stated:

- The strike of the option, which we will call “K”
- The market price of the underlying, which we will call “S.” S stands for “spot,” which for now we will just say is another name for the market price.
- The volatility of the spot price
- The time left until the option expires
- The “riskless rate” briefly mentioned above

K and S determine intrinsic value. The other variables deal more with what we call “time value.”

**Intrinsic Value versus Time Value**

Let’s consider a call option with a strike of $8 (K=$8). Let’s say the market price or spot price is $9 (S = $9). If you were to exercise at that moment, you would make exactly $1 in profit (intrinsic value = $1 because $9 – $8 = $1). This uses the same logic we went over above. If you don’t get it, try doing Question 1, asking for help in the comments, or DMing me.

But what if we knew that the price of the underlying (S) could change? For simplicity’s sake, we’ll assume for now that S can change exactly once before expiration. We’ll also assume that, when S does change, it will change to either $6 (go down by $3) or $12 (go up by $3), and there is a 50% chance of each of those possible changes happening. Finally, we’ll assume that the change is about to occur this instant, so no time will pass before the single change, and the change will happen instantly. This is a lot to assume, but the assumptions will become less ridiculous as we are variables to our model.

If S goes to $12, intrinsic value would become $4 because $12 – $8 = $4. If S goes down to $6, the call is worth $0. Why is it not worth $-2? Remember, a call option is the right, but **not the obligation** to buy the underlying at the strike, so the option holder does not need to buy the underlying for $8 when the spot price is at $6; they can just decide not to exercise the option and let it expire worthless.

Let’s recap: There is a 50% change the option will be worth $4 at expiration and a 50% chance it will be worth $0 at expiration. The option has $1 of intrinsic value right now. It also has what practitioners call “time value,” which is the value of the chance of the option having intrinsic value later. This time value is not to be confused with “the time value of money” which is a different concept.

Let’s see how much time value this option has, starting with the single most important fact about option valuation: **The total value of an option is a function of its probability weighted future payoffs.** In this case, to find that total value, you multiply the value of each outcome by the likelihood it will happen. 50%*$4 + 50%*$0 = $2, so the total value of the option is $2. Subtract the intrinsic value of $1 from the total value, and you get $1 of time value.

Before we do another example, this time with a put option, let’s quickly examine why it is worth breaking out the value of an option into intrinsic value versus time value. As I implied, **total option value = time value + intrinsic value**, and intrinsic value is S-K (spot – strike) for a call option and K-S (strike – spot) for a put option. You may recall I said that the one time an option’s total value is equal to its intrinsic value is at the moment of expiration because the underlying spot price has no time left to change. The implication here is that if total value approximates intrinsic value as the option nears expiration, then time value ultimately decreases to $0. Describing mathematically how the time value changes over time and eventually gets to $0 will require all five variables in either of our models – so that’s something to look forward to. For now, just be aware that this feature of options valuation exists.

Let’s take the simplified model we have already used to value a call and apply it to value a put option. I’m going to start using the variables without restating what they mean at this point. You will want to be comfortable with them by the time we get to a full model.

Consider this put: K=7. S=10. The spot can either become $3 or $17 using the same assumptions about changes in the spot that we made for the call option valuation above. Let’s check out the total valuation first. If the price goes down to $3, it will be $4 below K, so intrinsic value would be $4. If the price of the stock goes up to $17, the intrinsic value of the option would be $0 because $17>$7. Now, let’s add it up: 50%*$4 + 50%*$0 = $2, so the option is worth $2. How much of that is intrinsic value? Well, S>K to start with, so the option starts with no intrinsic value. Therefore, all $2 are from time value.

A little more terminology before we continue. **The relationship between a strike price and a spot price is called “moneyness.”** Remember how intrinsic value was $0 when the strike was below the spot for a put or above the spot for a call? **Intrinsic value cannot be negative, but moneyness can**. So a call with S = $3 and K = $5 would have a moneyness of $-2. If moneyness is positive, an option is said to be “in the money” or “ITM.” If you’re familiar with the musical *42nd Street*, you can remember this because being in the money is a very good thing. If moneyness is negative, the option is “out of the money”

or “OTM.” If moneyness is exactly $0, the option is “at the money” or “ATM.” This terminology will come in handy later when we start talking about actual trades. Let’s test our knowledge so far:

**Question 2:** An option has a moneyness of $-2 and S>K. Is it a call or a put? Answer shown at end of post. Can the option still be worth more than $0? How much of that value would be intrinsic value versus time value?

The relationship between moneyness and intrinsic value results in a special relationship between total option value and the spot price of the underlying.

Thank you for staying with me so far by the way. I know this is a lot.

**Volatility and Option Value**

You may have already noticed that an option’s intrinsic value can be expressed as a function of moneyness. Namely, if moneyness is at least 0, then intrinsic value = moneyness. This idea is powerful because it applies to both puts and calls.

Now consider that if moneyness is not greater than 0, intrinsic value does not change as moneyness decreases further because it just stays at 0. In other words, whether moneyness is -3 or -10,000, intrinsic value is 0. However, if moneyness is at least 0, then as moneyness increases, intrinsic value increases as we stated before. If we put these two ideas together, we see that **the relationship between moneyness and intrinsic value is asymmetrical around 0. In other words, options have an asymmetric payoff function.**

Now let’s think of this in terms of volatility: all else being equal, a more volatile underlying asset price has more potential to cause an option to expire further in the money or further out of the money. Another way of saying this is that greater volatility leads to a greater chance of a higher absolute value of moneyness at expiration. Since intrinsic value can’t get any lower once it hits 0, higher volatility can raise the maximum intrinsic value an option can have at expiration up to infinity, but cannot lower the minimum intrinsic value below 0. So, under the model we have been using, if an option can already end up with $0 of intrinsic value, volatility can only help option value, not hurt it, from there. Later, when we get to the Black Scholes model, we will see that volatility is virtually always is accretive to total option value, but for now we will stick to easy examples where options have a high potential to expire OTM.

For the purposes of an example, we’ll introduce one new term; σ (pronounced “sigma”). σ a parameter used to express volatility. Some readers may notice that σ is used to denote standard deviation in statistics. The idea is similar here. For this model, σ will denote the amount by which S can move up or down during the single change that occurs before option expiration. So if S is 0 to start with and can move to 4 or -4, then σ = 4. **Later on, we will express σ as a %, indicating a rate of change, but we are keeping it simple for now.**

Consider a call option with the following parameters using the model we have been using so far: K=25, S=20, σ=10. We know S can end up being either 30 (20+10) or 10 (20-10), which means the intrinsic value of the call will be either 5 (30-25) or 0 (because 10<25), which means that value of the option now is 2.5.

Now let’s try decreasing sigma to 7. Now S can now be either 27 or 13 at expiration, which means intrinsic value is going to end up being either 2 or 0, which means the option’s value is 1. Notice how lowering σ decreased total option value.

Now, let’s increase sigma dramatically to 15. S at expiration will either be 35 or 5, so intrinsic value at expiration with either be 10 or 0, which means the total value of the call option is 5.

Feel free to try this with as many puts and calls as you want. You will observe that as long as the option isn’t so far in the money that S-σ>K for a call or S+ σ<K for a put, an increase in σ can never decrease option value, and as long as S has the potential to change such that the option may expire OTM or ATM, an increase in volatility will always increase total option value.

**Congratulations: you now understand the basic idea of how options prices relate to the volatility of the underlying asset, and you can express it using a simple options pricing model. (It’s a simplified version of the “binomial model” for readers that want to put a name on it.)**

You may be wondering how markets determine volatility in the first place. When options trade on the market, the value that the options trade for can be used to determine the σ implied by the trade. For example, if we are using the model we’ve used above, a call option worth $3 where S starts at $5 and K is $4, must have an implied σ of $5. No other σ would produce that total option value.

In the real world, the implied level of volatility is typically expressed as a % and refers to the standard deviation of the change in S over the course of a year. The model used to calculate implied volatility is typically the Black Scholes model, which we can easily learn once we are able to use all 5 variables in the binomial model. The implied value of σ given an option’s price based on the Black Scholes model is called Implied Volatility or “IV”. We will deal with IV more later. For now, just think of it as the market’s best guess at σ.

The idea of IV does, however, bring up something we can use now; the IV of an option trading on the market is an estimation collectively arrived at by traders. **IV does not necessarily represent how much S will actually change because the market cannot psychically tell the future.** In other words, the market tries to calculate an appropriate value for σ, but often comes up with something that isn’t subsequently reflected in the actual behavior of the underlying. They get it “wrong,” in a way, and if you watch an option for a while, you’ll observe IV constantly changing.

There is a way to try to make money from this theory: **If you think σ based on the actual future behavior of the stock will be too high or too low to justify the value at which an option currently trades, you may have a trading opportunity.**

In that light, let’s do our first imaginary trade. To up the ante, **I’ll give gold to the first person who gets this right and posts the answer in the comments:** Let’s say you think the σ of an asset is going to be $10 based on the model we have used in the previous examples. You look at the option chain (the list of options available on the asset), and you see the options below. You decide you are going to buy one option. Which do you buy, and why?

S = $25

**Calls**

K = $35 Price = $1.00

K = $30 Price = $2.50

K = $25 Price = $3.50

K = $20 Price = $5.00

**Puts**

K = $30 Price = $3.50

K= $25 Price = $2.00

K = $20 Price = $1

K = $15 Price = $0.5

*Supplemental:*

*Bid Ask Spread:**The difference between the highest bid (offer to buy something) and the lowest ask (offer to sell something) is called the “bid ask spread.” The smaller the bid ask spread is, the more trades usually occur because small spreads imply there is less of a disagreement, measurable in price, between buyers and sellers. This small disagreement means only a small change in the bid or the ask is necessary to make a trade happen.*

*To Whom Do You Buy or Sell When You Exercise an Option?**Ultimately, financial responsibility will fall on the person who wrote (originated or originally sold) the option, but your broker and a group called a “clearing house” are the ones who ensure that a transaction actually occurs when you exercise your option. The important thing to remember is that you do not take counter party risk from the person who wrote the option. That is to say, the person who wrote the option can’t simply refuse to honor their agreement and cheat you. If they try, you’ll still get to profit (to the extent appropriate) when you exercise your option because of the clearing house. The writer becomes their broker’s problem (and eventually law enforcement’s problem) – not yours. Keep an eye out for a post about “assignment” or look up “assignment and clearing houses,” for further reading.*

*Answers to Practice Questions*

*Question 1: $5.You exercise to buy for $5, and then you sell in the market for $10.*

*Question 2: It’s a put, and it can be worth more than $0 if it has any time value. It has no intrinsic value.*

Disclaimer: This information is only for educational purposes. Do not make any investment decisions based on the information in this article. Do you own due diligence.